whether or not it goes through the centre of the earth. If you have a frictionless tunnel, so that the component of the force of gravity parallel to the tunnel is accelerating/decelerating you, while the rest is just a normal force at right angles, the time taken from one end to another of any tunnel is exactly the same - the maximum velocity is greatest when going through the centre of the earth, but so is the distance you need to cover.
No, I can't prove this any more (I remember it as a question back as school), but I think there will be a proof somewhere on the 'net.
On edit:
There's a related problem. In Chapter 7 of Lewis Carroll's 1893 book Sylvie and Bruno. The fictional German professor, Mein Herr, proposes a way to run trains by gravity alone. Dig a straight tunnel between any two points on Earth (it need not go through the Earth's center), and run a rail track through it. With frictionless tracks the energy gained by the train in the first half of the journey is equal to that required in the second half. And also, in the absence of air resistance and friction, the time of the journey is about 42 minutes (84 for a round trip) for any such tunnel, no matter what the tunnel's length.
Martin Gardner discussed this in one of his Scientific American "Mathematical Games" columns.
The period in the frictionless hole:
Restating the question: The period for the circular "skimming" orbit (of radius equal to the Earth's radius, R) is
T = 2p (R/g)1/2
Show that this is the same as the period of motion in frictionless straight hole through the center of the stationary Earth.
Partial Answer:
....
http://www.lhup.edu/~dsimanek/scenario/insight.htm
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